# 7.6 Techniques to Estimate Future Costs

## Scatter Graph Method to Calculate Future Costs at Varying Activity Levels

One of the assumptions that managers must make in order to use the cost equation is that the relationship between activity and costs is linear. In other words, costs rise in direct proportion to activity. A diagnostic tool that is used to verify this assumption is a scatter graph.

scatter graph shows plots of points that represent actual costs incurred for various levels of activity. Once the scatter graph is constructed, we draw a line (often referred to as a trend line) that appears to best fit the pattern of dots. Because the trend line is somewhat subjective, the scatter graph is often used as a preliminary tool to explore the possibility that the relationship between cost and activity is generally a linear relationship. When interpreting a scatter graph, it is important to remember that different people would likely draw different lines, which would lead to different estimations of fixed and variable costs. No one person’s line and cost estimates would necessarily be right or wrong compared to another; they would just be different. After using a scatter graph to determine whether cost and activity have a linear relationship, managers often move on to more precise processes for cost estimation, such as the high-low method or least-squares regression analysis.

To demonstrate how a company would use a scatter graph, let’s turn to the data for Regent Airlines, which operates a fleet of regional jets serving the northeast United States. The Federal Aviation Administration establishes guidelines for routine aircraft maintenance based upon the number of flight hours. As a result, Regent finds that its maintenance costs vary from month to month with the number of flight hours, as depicted in Table 7.2.

Table 7.2 Monthly Maintenance Cost and Activity Detail for Regent Airlines By: Rice University OpenStax CC BY-NC-SA 4.0
Month Activity Level (Flight Hours) Maintenance Costs
January 21,000 $84,000 February 23,000 90,000 March 14,000 70,500 April 17,000 75,000 May 10,000 64,500 June 19,000 78,000 When creating the scatter graph, each point will represent a pair of activity and cost values. Maintenance costs are plotted on the vertical axis (Y), while flight hours are plotted on the horizontal axis (X). For instance, one point will represent 21,000 hours and$84,000 in costs. The next point on the graph will represent 23,000 hours and $90,000 in costs, and so forth, until all of the pairs of data have been plotted. Finally, a trend line is added to the chart in order to assist managers in seeing if there is a positive, negative, or zero relationship between the activity level and cost. Figure 7.59 shows a scatter graph for Regent Airlines. In scatter graphs, cost is considered the dependent variable because cost depends upon the level of activity. The activity is considered the independent variable since it is the cause of the variation in costs. Regent’s scatter graph shows a positive relationship between flight hours and maintenance costs because, as flight hours increase, maintenance costs also increase. This is referred to as a positive linear relationship or a linear cost behavior. Will all cost and activity relationships be linear? Only when there is a relationship between the activity and that particular cost. What if, instead, the cost of snow removal for the runways is plotted against flight hours? Suppose the snow removal costs are as listed in Table 7.3. Table 7.3 Scatter Graph of Snow Removal Costs for Regent Airlines. By: Rice University OpenStax CC BY-NC-SA 4.0 Month Activity Level: Flight Hours Snow Removal Costs January 21,000$40,000
February 23,000 50,000
March 14,000 8,000
April 17,000 0
May 10,000 0
June 19,000 0

As you can see from the scatter graph, there is really not a linear relationship between how many flight hours are flown and the costs of snow removal. This makes sense as snow removal costs are linked to the amount of snow and the number of flights taking off and landing but not to how many hours the planes fly.

Using a scatter graph to determine if this linear relationship exists is an essential first step in cost behavior analysis. If the scatter graph reveals a linear cost behavior, then managers can proceed with a more sophisticated analyses to separate mixed costs into their fixed and variable components. However, if this linear relationship is not present, then other methods of analysis are not appropriate. Let’s examine the cost data from Regent Airline using the high-low method.

## High-Low Method to Calculate Future Costs at Varying Activity Levels

As you’ve learned, the purpose of identifying costs is to control them, and managers regularly use past costs to predict future costs. Since we know that variable costs change with the level of activity, we can conclude that there is usually a positive relationship between cost and activity: As one rises, so does the other. Ideally, this can be confirmed on a scatter graph. One of the simplest ways to analyze costs is to use the high-low method, a technique for separating the fixed and variable cost components of mixed costs. Using the highest and lowest levels of activity and their associated costs, we are able to estimate the variable cost components of mixed costs.

Once we have established that there is linear cost behavior, we can equate variable costs with the slope of the line, expressed as the rise of the line over the run. The steeper the slope of the line, the faster costs rise in response to a change in activity. Recall from the scatter graph that costs are the dependent Y variable and activity is the independent X variable. By examining the change in Y relative to the change in X, we can predict cost:

$Variable\;Cost=\frac{Rise\;of\;the\;line}{Run\;of\;the\;line}=\frac{y_2-y_1}{x_2-x_1}$

where Y2 is the total cost at the highest level of activity; Y1 is the total cost at the lowest level of activity; X2 is the number of units, labor hours, etc., at the highest level of activity; and X1 is the number of units, labor hours, etc., at the lowest level of activity.

Using the maintenance cost data from Regent Airlines shown in Figure 7.60, we will examine how this method works in practice.

Table 7.4 Monthly Maintenance Cost and Activity Detail for Regent Airlines. By: Rice University OpenStax CC BY-NC-SA-4.0
Month Activity Level (Flight Hours) Maintenance Costs
January 21,000 $84,000 February 23,000 90,000 March 14,000 70,500 April 17,000 75,000 May 10,000 64,500 June 19,000 78,000 The first step in analyzing mixed costs with the high-low method is to identify the periods with the highest and lowest levels of activity. In this case, it would be February and May, as shown in Figure 7.60. We always choose the highest and lowest activity and the costs that correspond with those levels of activity, even if they are not the highest and lowest costs. Table 7.5 High–Low Data Points for Regent Airlines Maintenance Costs. By: Rice University OpenStax CC BY-NC-SA 4.0 High/Low Activity Level(Flight Hours) Cost (Maintenance Costs) Highest level (February) 23,000$90,000
Lowest level (May) 10,000 64,500

We are now able to estimate the variable costs by dividing the difference between the costs of the high and the low periods by the change in activity using this formula:

$Variable\;Cost=\frac{Change\;in\;Cost}{Change\;in\;Activity}=\frac{\text {Cost at the high activity level}-\text {Cost at the low activity level}}{\text {Highest activity level}-\text{Lowest activity level}}$

For Regent Airlines, this is:

$Variable\;Cost=\frac{\90,000-\64,500}{23,000=10,000}=\1.96\;per\;flight\;hour$

Having determined that the variable cost per flight-hour is $1.96, we can now determine the amount of fixed costs. We can determine these fixed costs by taking the total costs at either the high or the low level of activity and subtracting this variable component. You will recall that total cost = fixed costs + variable costs, so the fixed cost component for Regent Airlines can be isolated as shown: $\begin{array}{c}\text{Fixed cost}=\text{total cost}–\text{variable cost}\hfill \\ \text{Fixed cost}=90,000–\left(23,000\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}1.96\right)\hfill \\ \text{Fixed cost}=44,920\hfill \end{array}$ Notice that if we had chosen the other data point, the low cost and activity, we would still get the same fixed cost of$44,920 = [$64,500 – (10,000 ×$1.96)].

Now that we have isolated both the fixed and the variable components, we can express Regent Airlines’ cost of maintenance using the total cost equation:

$Y=\44,920+\1.96x$

where Y is total cost and x is flight hours.

Because we confirmed that the relationship between cost and activity at Regent exhibits linear cost behavior on the scatter graph, this equation allows managers at Regent Airlines to conclude that for every one unit increase in activity, there will be a corresponding rise in variable cost of $1.96. When put into practice, the managers at Regent Airlines can now predict their total costs at any level of activity, as shown in Figure 7.60. Table 7.6 Predictions of Total Cost and Cost Components at Different Levels of Activity for Regent Airlines. By: Rice University OpenStax CC BY-NC-SA 4.0 Activity Level (Flight Hours) Fixed Costs Variable Cost at$1.96 per hour Total Costs
10,000 $44,920$19,600 \$64,520
20,000 44,920 39,200 84,120
30,000 44,920 58,800 103,720
40,000 44,920 78,400 123,320

Although managers frequently use this method, it is not the most accurate approach to predicting future costs because it is based on only two pieces of cost data: the highest and the lowest levels of activity. Actual costs can vary significantly from these estimates, especially when the high or low activity levels are not representative of the usual level of activity within the business. For a more accurate model, the least-squares regression method would be used to separate mixed costs into their fixed and variable components. The least-squares regression method is a statistical technique that may be used to estimate the total cost at the given level of activity based on past cost data. Least-squares regression minimizes the errors of trying to fit a line between the data points and thus fits the line more closely to all the data points.